Welcome to the forums
My first thought is that you may have introduced the cats too early. Did you keep your wife's cat confined for a while before introducing them? If so, how long for? Did you give them a very slow introduction, such as exchanging their scents by rubbing them with blankets scented by the other cats, did you let them sniff each other through the door? Cats are extremely territorial and an invader can cause extreme distress, particularly if the cats feel they don't have a 'safe zone'.
The first thing I would try is to 'reintroduce' the animals. Confine your wife's cat to one room for a few days, don't let the other cats in. Let your two cats get used to having the house back to themselves. Give them lots of reassurance - treats and attention, play with them a lot, show them the only thing changing right now is a change for the better. Let them sniff at the door of the room your wife's cat is being confined in, but don't let them fixate - distract them with treats and toys.
After a few days when they have settled down a bit, start scent swapping. Associate the new cats scent with positive things. Put a blanket that has been rubbed on your wife's cat by their food bowls. Put the new-cat scented blanket down and have them play with toys over the top of it. Make sure to do the same with your wife's cat, and make sure your wife's cat is getting plenty of attention - I'm sure the cat is not used to being confined, and not used to being in a new house. She needs a lot of reassurance right now, just as your other cats do. Make sure you and your wife pay lots of attention to both sets of cats so they know things are going to be okay in this new living situation.
When your cats can sniff at the door without growling and hissing, you might open the door a crack and see how they go with scenting each other. You can let them hiss and growl and get it all out, but if they seem particularly agitated (raised fur, spitting) close the door, give them attention (play) and try again later. You might want to try leaving the door open a sliver so they can see each other, but playing with your two cats so they get distracted. You want to get them relaxed in the presence of the new cat.
After a few introductions they shouldn't be TOO phased about the new cat (spitting should have abated, short, half-hearted hisses are okay) and you should be able to let your cats into the room. It's best to let your cats into the room your wife's cat has been in as your cats are the ones who are going to feel most territorial, so they will be less bothered if they get to explore new territory instead of a new cat invading theirs.
Make sure to have one-on-one introductions at first, and try not to have the cats interact - you and your wife should each play with one, so that the cats are in the same room, getting used to each others scent, but not having time to get upset about the others presence. When they seem calm in each others presence (happy to play and not keep an eye on the other one every 20 seconds) it should be okay to let them near each other and let them sniff. Hissing is fine. Don't ever yell at them while they are interacting or make them associate the other with a negative reaction. If things seem like they're going to get hairy then clap loudly, bang on something or distract with a toy, try not to directly interfere.
When the cats seem okay with each other, it should be okay to let the new cat out into the house, with both of the household cats in attendance. By then your cats should be used to the new kitty and be able to tolerate its presence and not be so distressed.
In the meantime, while reintroduction is occurring, you might want to look into getting a Feliway diffuser to calm your cats.
I know that having a cat urinate everywhere and on everything is an extremely taxing, stressful experience, but please remember that part of being a responsible pet owner is making sure your pet is as comfortable and unthreatened as possible. Your urinating cat is simply scared and upset, due to factors beyond her control. She probably isn't feeling safe right now. The problem might be exacerbated if you were to give her up, as some owners might do when confronted with such a difficult situation - I've heard too many horror stories about cats with stress-related urination issues being put down because no one took the time to eliminate the root of the problem.
Obviously the fact that your posting here asking for help means you really do love your kitty and want to help her, so that's really great. I really believe you can fix this with just a bit of time and patience.
You also might want to look into clicker-training - it's a very easy, quick and effective way to train cats, and has done wonders for some who have had to deal with the same issue you're dealing with right now.
I'm sure there is a lot of other great advice to be given, but this post has gone on too long and I'm far from an expert, so I'll leave that up to someone else. Feel free to PM me if you have any questions